I gave it a try with the gmail settings but get the following error when I clicked 'Test mail settings':

Could not connect - Request for the permission of type 'System.Net.Mail.SmtpPermission, System, Version=4.0.0.0, Culture=neutral, PublicKeyToken=b77a5c561934e089' failed.

Just some general information, I am hosting on GoDaddy using BE 1.6.1. I entered my gmail address in the e-mail and username, password and smtp.gmail.com. I get this error for both ports 465 and 587. So I am either doing something
wrong. Or there is a bug in BE.

I'm almost certain there are more error details that are not being displayed (and they are the more important error details). In the Settings.aspx.cs page in the Admin\Pages folder, you'll find the code that displays this error message you are seeing.
It's inside of "btnTestSmtp_Click". In there is a "catch" block of code. Modify it so it looks like this:

Ben, forgive me for jumping into someone else's thread but I've had the same issue. I made the change that you suggested and I still only get the
Could not connect - Failure sending mail. that billyhum received.

For what it worth, It's been a while and might changed since... When I was on GoDaddy I had to use their email account to make SMTP work. (I then set redirect to forward all to gmail). Some details
here.

I apologize to all, that I did not post this sooner. I did not receive any new response to my previous comment about it still not working at my email that I forgot all about it. I did find a solution a couple of weeks ago. And what rtur says is correct.
GoDaddy does not allow for someone to use another SMTP through their hosting environment. What you need to do is setup an email account on GoDaddy. Then you use their smtp server to do the email using the email account you set up. You can find out more about
this by doing a search for 'smtp server' through GoDaddy's support community.